About the twin prime number conjecture

Question

Let $\pi (x)$ be the number of prime numbers less than $x$. If $p-q=2k$ for fixed $k$, then:

$$\pi^{-1}(n)-\pi^{-1}(m)=2k$$

We apply Taylor development:

$$\frac{1}{\pi' \circ \pi^{-1} (c)} (n-m)=2k$$

But $\pi (x)\sim li(x)$, $li (x)=\int_2^x \frac{dt}{ln(t)}$ the integral logarithm of Gauss.

$$ln(\pi^{-1}(c))\sim \frac{2k}{(n-m)}$$

$$c \leq \pi(e^{2k})$$

Thus we deduce the twin prime number conjecture for $k=1$. 

Answer 1

Badly, the formula $\pi (x)= li(x)+O(x^{1/2+\epsilon})$ seems to be false. We have to regularize the sum

$$\sum_{\rho} li(x^{\rho})$$

where $\rho$ are the zeros of the Riemann zeta function.